Complex number

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TOPICS :---

‘‘Complex Number”

History.

Number System.

Complex numbers.

Operations.

5(Complex Number)

Contents

Complex numbers were first introduced by G. Cardano

R. Bombelli introduced the symbol .

A. Girard called “solutions impossible”.

C. F. Gauss called “complex number” 6(Complex Number)

History

7(Complex Number)

Number System

Real Number

Irrational

Number

Rational

Number

Natural Numbe

r

Whole Numbe

rInteger

Imaginary

Numbers

8(Complex Number)

Complex Numbers

• A complex number is a number that can b express in the form of

• Where a and b are real number and is an imaginary.

• In this expression, a is the real part and b is the imaginary part of complex number.

Complex Number

Real Numbe

r

Imaginary

Number

Complex

Number

When we combine the real and imaginary number then complex number is form.

10(Complex Number)

Complex Number• A complex number has a real part and an

imaginary part, But either part can be 0 .• So, all real number and Imaginary number are

also complex number.

11(Complex Number)

Complex NumbersComplex number convert our visualization into physical things.

A complex number is a number consisting of a Real and Imaginary part. It can be written in the form

COMPLEX NUMBERS

1i

COMPLEX NUMBERS Why complex numbers are

introduced??? Equations like x2=-1 do not have a solution within the real numbers12 x

1x

1i

12 i

COMPLEX CONJUGATE

The COMPLEX CONJUGATE of a complex number

z = x + iy, denoted by z* , is given by

z* = x – iy The Modulus or absolute value is defined by

22 yxz

COMPLEX NUMBERS

Equal complex numbers

Two complex numbers are equal if theirreal parts are equal and their imaginaryparts are equal.

If a + bi = c + di, then a = c and b = d

idbcadicbia )()()()(

ADDITION OF COMPLEX NUMBERS

iii

)53()12()51()32(

i83

EXAMPLE

Real Axis

Imaginary Axis

1z

2z

2z

sumz

SUBTRACTION OF COMPLEX NUMBERS

idbcadicbia )()()()(

ii

ii

21)53()12()51()32(

Example

Real Axis

Imaginary Axis

1z

2z

2z

diffz

2z

MULTIPLICATION OF COMPLEX NUMBERS

ibcadbdacdicbia )()())((

ii

ii

1313)310()152(

)51)(32(

Example

DIVISION OF A COMPLEX NUMBERS

dic

bia

dic

dicdicbia

22

2

dcbdibciadiac

22 dc

iadbcbdac

EXAMPLE

i

i2176

i

iii

2121

2176

22

2

21147126

iii

415146

i

5520 i

55

520 i

i4

DE MOIVRE'S THEORoM

DE MOIVRE'S THEORM is the theorm which show us how to take complex number to any power easily.

Euler Formula

jre

jyxjrz

)sin(cos

yjye

eeee

jyxz

x

jyxjyxz

sincos

This leads to the complex exponential function :

The polar form of a complex number can be rewritten as

So any complex number, x + iy, can be written inpolar form:

Expressing Complex Number in Polar Form

sinry cosrx

irryix sincos

A complex number, z = 1 - j has a magnitude

2)11(|| 22 z

Example

rad24

211tan 1

nnzand argument :

Hence its principal argument is :

rad

Hence in polar form :

4

zArg

4sin

4cos22 4

jezj

EXPRESSING COMPLEX NUMBERS IN POLAR FORM

x = r cos 0 y = r sin 0

Z = r ( cos 0 + i sin 0 )

APPLICATIONS

Complex numbers has a wide range of applications in Science, Engineering, Statistics etc. Applied mathematics Solving diff eqs with function of complex roots

Cauchy's integral formula

Calculus of residues

In Electric circuits to solve electric circuits

Examples of the application of complex numbers:

1) Electric field and magnetic field.2) Application in ohms law.

3) In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes

4) A complex number could be used to represent the position of an object in a two dimensional plane,

How complex numbers can be applied to “The Real World”???

REFERENCES..

Wikipedia.com Howstuffworks.com Advanced

Engineering Mathematics

Complex Analysis